Theorem idlcl 39058

Description: Obsolete theorem, use 2idllidld 21554 and lidlbasel 21498 instead. An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
idlss.1	⊢ 𝐺 = (1st ‘𝑅)
idlss.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
idlcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)

Proof of Theorem idlcl

Step	Hyp	Ref	Expression
1		idlss.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		idlss.2	. . 3 ⊢ 𝑋 = ran 𝐺
3	1, 2	idlss 39057	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	sselda 3909	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ran crn 5651  ‘cfv 6541  1^st c1st 8001  RingOpscrngo 38935  Idlcidl 39048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6496  df-fun 6543  df-fv 6549  df-ov 7425  df-idl 39051

This theorem is used by:  idlsubcl  39064